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January 8th, 2017, 12:34 PM   #1
kko
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optimization problem

Consider an optimization problem,

minimize f(Ax)+x^T x

the variable is n-vector x. The matrix A has size m by n and rank m. f is not necessarily differentiable or convex.

Show that the problem can be formulated as an equivalent problem with m variables, by making change of y = Ax.
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January 26th, 2017, 05:30 AM   #2
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Since $\displaystyle y = Ax, x = A^{-1} y$,

$\displaystyle A^{-1} A = 1$

and

$\displaystyle (AB)^T = B^T A^T$

$\displaystyle f(Ax) + x^T x$
$\displaystyle = f(y) + (A^{-1} y)^T A^{-1} y$
$\displaystyle = f(y) + y^T (A^{-1})^T A^{-1} y$


This is almost exactly the same as the original equation except that y is the new variable of interest and you have to work out $\displaystyle (A^{-1})^T A^{-1}$ as the filling of the matrix sandwich

Last edited by Benit13; January 26th, 2017 at 05:41 AM.
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